Where this approach has difficulty is in understanding how God can have knowledge of changing things and himself not be changing -- if the thing known changes, doesn't it imply that God must also change, at least in his knowledge?

It seems to me, though, that this difficulty is illusory. If God is not subject to any change, then He exists outside of space-time. This means God doesn't know the things He knows as changing things; He doesn't observe things while they change, since He experiences no "while."

A common analogy is that we can look at a piece of paper and see an unchanging rectangle, which a sentient point on the paper, traveling around the rectangle, can only see a line segment of varying length (and intensity, if like Flatland, objects are luminous and there's a bit of fog about).

Now, this analogy is in effect one of the "science of vision," whereby God "sees all things in His eternal Present." That may or may not be true of God, although I'm with St. Thomas in thinking it is; the important point is that the existence of changing things is not incompatible with the existence of an unchanging God Who, in one way or another, knows the changing things.